Transcript WISSystematics18Dec08 - University of Toronto, Particle Physics

Systematic Uncertainties:
Principle and Practice
Outline
1. Introduction to Systematic Uncertainties
2. Taxonomy and Case Studies
3. Issues Around Systematics
4. The Statistics of Systematics
5. Summary
Pekka K. Sinervo,F.R.S.C.
Rosi & Max Varon Visiting Professor
Weizmann Institute of Science
&
Department of Physics
University of Toronto
18 Dec 08
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Introduction

Systematic uncertainties play key role in physics
measurements
– Few formal definitions exist, much “oral tradition”
– “Know” they are different from statistical uncertainties
Random Uncertainties




Arise from stochastic
fluctuations
Uncorrelated with previous
measurements
Well-developed theory
Examples



measurement resolution
finite statistics
random variations in system
Systematic Uncertainties




Due to uncertainties in the
apparatus or model
Usually correlated with
previous measurements
Limited theoretical framework
Examples



calibrations uncertainties
detector acceptance
poorly-known theoretical
parameters
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Literature Summary

Increasing literature on the topic of “systematics”
A representative list:
–
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R.D.Cousins & V.L. Highland, NIM A320, 331 (1992).
C. Guinti, Phys. Rev. D 59 (1999), 113009.
G. Feldman, “Multiple measurements and parameters in the unified approach,”
presented at the FNAL workshop on Confidence Limits (Mar 2000).
R. J. Barlow, “Systematic Errors, Fact and Fiction,” hep-ex/0207026 (Jun 2002), and
several other presentations in the Durham conference.
G. Zech, “Frequentist and Bayesian Confidence Limits,” Eur. Phys. J, C4:12 (2002).
R. J. Barlow, “Asymmetric Systematic Errors,” hep-ph/0306138 (June 2003).
A. G. Kim et al., “Effects of Systematic Uncertainties on the Determination of
Cosmological Parameters,” astro-ph/0304509 (April 2003).
J. Conrad et al., “Including Systematic Uncertainties in Confidence Interval
Construction for Poisson Statistics,” Phys. Rev. D 67 (2003), 012002
G.C.Hill, “Comment on “Including Systematic Uncertainties in Confidence Interval
Construction for Poisson Statistics”,” Phys. Rev. D 67 (2003), 118101.
G. Punzi, “Including Systematic Uncertainties in Confidence Limits”, CDF Note in
preparation.
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I. Case Study #1: W Boson Cross
Section

Rate of W boson production
– Count candidates Ns+Nb
– Estimate background
Nb & signal efficiency e
  N c  N b  (e L)
– Measurement reported as

  2.64  0.01 (stat)
 0.18 (syst) nb
– Uncertainties are


 stat   0 1/N c
 syst   0 N b /N b   e /e  L /L
2
2
2
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Definitions are Relative

Efficiency uncertainty estimated using Z
boson decays
– Count up number of Z candidates NZcand
 Can identify using charged tracks
 Count up number reconstructed NZrecon
recon
N
e  Z cand  e 
NZ
NZ
recon
N
cand
Z
 NZ
N Z cand
– Redefine uncertainties
2
 stat   0 1/N c  e /e
–
2
2
 syst   0 N b /N b   L /L


recon

Lessons:
• Some systematic uncertainties
are really “random”
• Good to know this
• Uncorrelated
• Know how they scale
• May wish to redefine
• Call these
“CLASS 1” Systematics
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Top Mass Good Example

Top mass uncertainty in template analysis
– Statistical uncertainty from shape of
reconstructed mass distribution and
statistics of sample
– Systematic uncertainty coming from jet
energy scale (JES)
 Determined by calibration studies,
dominated by modelling uncertainties
 5% systematic uncertainty

Latest techniques determine JES
uncertainty from dijet mass peak (W->jj)
– Turn JES uncertainty into a largely
statistical one
– Introduce other smaller systematics
M top  171.8  1.9(stat + JES)  1.0 (syst) GeV/c 2
 171.9  2.1 GeV/c 2
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Case Study #2: Background
Uncertainty

Look at same W cross section analysis
– Estimate of Nb dominated by QCD backgrounds
 Candidate event
– Have non-isolated leptons
– Less missing energy


Assume that isolation
and MET uncorrelated
Have to estimate the
uncertainty on NbQCD
– No direct measurement
has been made to verify the model
– Estimates using Monte Carlo modelling have large
uncertainties
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Estimation of Uncertainty

Fundamentally different class of uncertainty
– Assumed a model for data interpretation
– Uncertainty in NbQCD depends on accuracy of model
– Use “informed judgment” to place bounds on one’s
ignorance
 Vary the model assumption to estimate robustness
 Compare with other methods of estimation

Difficult to quantify in consistent manner
– Largest possible variation?
 Asymmetric?
– Estimate a “1 ” interval?

– Take  
?
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Lessons:
• Some systematic uncertainties
reflect ignorance of one’s data
• Cannot be constrained by
observations
• Call these
“CLASS 2” Systematics
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Case Study #3: Boomerang CMB
Analysis

Boomerang is one of several
CMB probes
– Mapped CMB anisoptropy
– Data constrain models of the
early universe

Analysis chain:
– Produce a power spectrum for
the CMB spatial anisotropy
 Remove instrumental effects through a complex
signal processing algorithm
– Interpret data in context of many models with
unknown parameters
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Incorporation of Model
Uncertainties

Power spectrum extraction
includes all instrumental
effects
– Effective size of beam
– Variations in data-taking
procedures

Use these data to extract
7 cosmological parameters
– Take Bayesian approach
 Family of theoretical models defined by 7 parameters
 Define a 6-D grid (6.4M points), and calculate likelihood
function for each
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Marginalize Posterior Probabilities

Perform a Bayesian
“averaging” over a grid
of parameter values
– Marginalize w.r.t. the
other parameters

NB: instrumental
uncertainies included
in approximate manner
– Chose various priors
in the parameters

Comments:
– Purely Bayesian analysis with
no frequentist analogue
– Provides path for inclusion of
additional data (eg. WMAP)
Lessons:
• Some systematic uncertainties
reflect paradigm uncertainties
• No relevant concept of a
frequentist ensemble
• Call these
“CLASS 3” Systematics
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Proposed Taxonomy for Systematic
Uncertainties

Three “classes” of systematic uncertainties
– Uncertainties that can be constrained by ancillary
measurements
– Uncertainties arising from model assumptions or
problems with the data that are poorly understood
– Uncertainties in the underlying models

Estimation of Class 1 uncertainties straightforward
– Class 2 and 3 uncertainties present unique challenges
– In many cases, have nothing to do with statistical
uncertainties
 Driven by our desire to make inferences from the data
using specific models
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II. Estimation Techniques


No formal guidance on how to define a systematic
uncertainty
– Can identify a possible source of uncertainty
– Many different approaches to estimate their magnitude

 Determine maximum effect 
 ?
2
General rule:


?
– Maintain consistency with definition of
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statistical intervals
– Field is pretty glued to 68% confidence intervals
– Recommend attempting to reflect that
 in magnitudes of
systematic uncertainties
– Avoid tendency to be “conservative”
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Estimate of Background
Uncertainty in Case Study #2

Look at correlation of Isolation and MET
– Background estimate
increases as isolation
“cut” is raised
– Difficult to measure or
accurately model
 Background comes
primarily from very
rare jet events with
unusual properties
 Very model-dependent

Assume a systematic uncertainty representing
the observed variation
– Authors argue this is a “conservative” choice
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Cross-Checks Vs Systematics

R. Barlow makes the point in Durham(PhysStat02)
– A cross-check for robustness is not an invitation to introduce
a systematic uncertainty
 Most cross-checks confirm that interval or limit is robust,
– They are usually not designed to measure a systematic
uncertainty

More generally, a systematic uncertainty should
– Be based on a hypothesis or model with clearly stated
assumptions
– Be estimated using a well-defined methodology
– Be introduced a posteriori only when all else has failed
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III. Statistics of Systematic
Uncertainties

Goal has been to incorporate systematic uncertainties
into measurements in coherent manner
– Increasing awareness of need for consistent practice
 Frequentists: interval estimation increasingly sophisticated
– Neyman construction, ordering strategies, coverage properties

Bayesians: understanding of priors and use of posteriors
– Objective vs subjective approaches, marginalization/conditioning
– Systematic uncertainties threaten to dominate as precision
and sensitivity of experiments increase

There are a number of approaches widely used
– Summarize and give a few examples
– Place it in context of traditional statistical concepts
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Formal Statement of the Problem

Have a set of observations xi, i=1,n
– Associated probability distribution function (pdf) and
likelihood function
p x |q  L q 
p x |q



i

  i 
i

Depends on unknown random parameter q
Have some additional uncertainty in pdf

– Introduce a second unknown parameter l
L q, l   px i | q, l
i

In some cases, one can identify statistic yj that
provides information about l

L q, l    px i , y j | q, l 
i, j
– Can treat l as a “nuisance parameter”

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Bayesian Approach

Identify a prior pl for the “nuisance parameter” l
– Typically, parametrize as either a Gaussian pdf or a flat
distribution within a range (“tophat”)
– Can then define Bayesian posterior
L q, l  p l dq dl
– Can marginalize over possible values of l
 Use marginalized posterior to set Bayesian credibility
intervals,
estimate parameters, etc.


Theoretically straightforward ….
– Issues come down to choice of priors for both q,l
 No widely-adopted single choice
 Results have to be reported and compared carefully to
ensure consistent treatment
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Frequentist Approach

Start with a pdf for data px i, y j | q, l
– In principle, this would describe frequency
distributions of data in multi-dimensional space
– Challenge is take account
of nuisance parameter

– Consider a toy model
px, y | ,n   Gx    n ,1Gy  n ,s



Parameter s is Gaussian
width for n
Likelihood function (x=10, y=5)
– Shows the correlation
– Effect of unknown n
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Formal Methods to Eliminate
Nuisance Parameters

Number of formal methods exist to eliminate
nuisance parameters
– Of limited applicability given the restrictions
– Our “toy example” is one such case
 Replace x with t=x-y and parameter n with
2

s
v' n 
1 s2


2
2
ts
s
 pt, y | ,n '  G t  , 1 s Gy  n '

2,
1
s

1 s2 


Factorized pdf and can now integrate over n’
 Note that pdf for  has larger width, as expected
– In practice, one often loses information using this
technique


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Alternative Techniques for
Treating Nuisance Parameters

Project Neyman volumes onto parameter of
interest
– “Conservative interval”
– Typically over-covers,
possibly badly

Choose best estimate of
nuisance parameter
– Known as “profile method”
– Coverage properties
From G. Zech
require definition of ensemble
– Can possible under-cover when parameters strongly
correlated
 Feldman-Cousins intervals tend to over-cover slightly
(private communication)
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Example: Solar Neutrino Global
Analysis

Many experiments have measured solar neutrino flux
– Gallex, SuperKamiokande, SNO, Homestake, SAGE, etc.
– Standard Solar Model (SSM) describes n spectrum
– Numerous “global analyses” that synthesize these

Fogli et al. have detailed one such analysis
– 81 observables from these experiments
– Characterize systematic uncertainties through 31 parameters
 12 describing SSM spectrum
 11 (SK) and 7 (SNO) systematic uncertainties

Perform a c2 analysis
– Look at c2 to set limits on parameters
Hep-ph/0206162, 18 Jun 2002
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Formulation of c2

In formulating c2, linearize effects of the systematic
uncertainties on data and theory comparison
 
 R exp t  R theor   (c kx )
n
n
n k
N


c 2pull  min x 
un
n1 
 
 


2
 K 


2
  x k 
k1






Uncertainties un for each observable
– Introduce “random” pull xk for each systematic
 Coefficients ckn to parameterize effect on nth observable
 Minimize c2 with respect to xk
 Look at contours of equal  c2

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Solar Neutrino Results

Can look at “pulls” at c2
minimum
– Have reasonable distribution
– Demonstrates consistency of
model with the various
measurements
– Can also separate
 Agreement with experiments
 Agreement with systematic
uncertainties
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Pull Distributions for Systematics

Pull distributions for xk
also informative
– Unreasonably small variations
– Estimates are globally too
conservative?
– Choice of central values
affected by data
 Note this is NOT a
blind analysis

But it gives us some
confidence that intervals
are realistic
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Typical Solar Neutrino Contours

Can look at probability
contours
– Assume standard c2 form
– Probably very small
probability contours have
relatively large
uncertainties
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Hybrid Techniques

A popular technique (Cousins-Highland) does an
“averaging” of the pdf
– Assume a pdf for nuisance parameter g(l)
– “Average” the pdf for data x
pCH x | q 
 px | q, lgl dl
– Argue this approximates an ensemble where
 Each measurement uses an apparatus that differs in

parameter l
– The pdf g(l) describes the frequency distribution


Resulting distribution for x reflects variations in l
Intuitively appealing
See, for example, J. Conrad et al.
– But fundamentally a Bayesian approach
– Coverage is not well-defined
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Summary

HEP & Astrophysics becoming increasingly
“systematic” about systematics
– Recommend classification to facilitate understanding
 Creates more consistent framework for definitions
 Better indicates where to improve experiments
– Avoid some of the common analysis mistakes
 Make consistent estimation of uncertainties
 Don’t confuse cross-checks with systematic uncertainties

Systematics naturally treated in Bayesian framework
– Choice of priors still somewhat challenging

Frequentist treatments are less well-understood
– Challenge to avoid loss of information
– Approximate methods exist, but probably leave the “true
frequentist” unsatisfied
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